WHAT DOES HAVE TO SAY ABOUT THE FOUNDATIONS OF

COMPUTER SCIENCE?

D. E. STEVENSON

Abstract. It is fair to say that Euclid's Elements has b een a driving factor in the developmentof

mathematics and mathematical logic for twenty-three centuries. The author's own love a air with

mathematics and logic start with the Elements.

But who was the man? Do we know anything ab out his life? What ab out his times? The

contemp orary view of Euclid is much di erent than the man presented in older histories. And what

would Euclid say ab out the status of computer science with its rules ab out everything?

1. Motivations

Many older mayhave started their love a air with mathematics with a high

scho ol course in plane geometry. Until recently, plane geometry was taught from translations of the

text suchasby Heath [5]. In myown case, it was the neat, clean picture presented with its neat,

clean pro ofs. I would guess that over the twenty-three hundred year history of the text, Euclid

must be the all-time, b est selling mathematics textb o ok author. At the turn of the last century,

David Hilb ert also found a love of Euclid. So much so that Hilb ert was moved to prop ose his

Formalist program for mathematics. Grossly over-simpli ed, Hilb ert prop osed that mathematics

was a \meaningless game" in that the rules of logic preordained the results indep endent of any

interpretation.

Hilb ert was rather vehemently opp osed by L. E. J. Brouwer. Brouwer prop osed that it was

the intuition of the who used a p ersonal language to develop mathematics; this

language always has meaning [14 ]. A fo cus of Brouwer was the concept of existence. Pretending to

1991 Mathematics Subject Classi cation. 14A99. 1

use an ob ject that has not b een constructed is called impredication. Henr Poincar e said impredi-

cation was the ro ot of all mathematical evil. Hilb ert's program eventually would fail, done in by

Godel's Incompleteness Theorem. But by no means did this establish intuitionism as the wave of

the future.

Intuitionism has several facets; it is the existence no impredication facet that broughtmeto

study absolute geometry part of Bo ok I with an eye to the history and philosophy of mathematics

in Euclidean times. The cause for this was an article byvon [15 ] who prop osed constructive

1

axioms for geometry Brie y, the di erence between constructive and platonic can be stated in

terms of Plato and his student Aristotle. This story is well-known in philosophy. Hop efully, not

oversimplifying, the question is over existence.

Platonic mathematics p ostulates that things just exist. For example, the uncountable set of

real numb ers exists as an ideal and we can treat them as if we had access to this set. Essentially,

this is a statement of Plato's philosophy. Plato's most famous pupil, Aristotle, disagreed, saying

that things must be shown to exist by some metho d or other. Aristotle's stand is often called

\empiricism." In mo dern garb, the argument was between Hilb ert's formalism and Brouwer's

intuitionism. It is said that \All mathematicians are intuitionists on Sunday and Platonist the rest

of the week." Errett Bishop is the most recent prop onent[1]. I nd the intuitionistic argument far

more satisfying in computer science: you cannot compute with something until you have created

it.

According to the the Platonist, Euclid was a platonic mathematician | go gure. This view

was p opularized primarily by Pro clus who once headed Plato's Academy and wrote an extensive

commentary on the Elements[5 ]. Unfortunately, Pro clus lived ab out six hundred years after Euclid.

The history of the platonic versus constructionist views are well do cumented in [2 , 6 ] and the details

1

I use the terms intuitionism and constructivism interchangeably. This may not b e technically correct but the

term construction is an inherent part of the geometric discussion 2

Alexander of Aphro disias Amphinomus Anaxgoras

Antiphon Aristotle Bryson

Demo citus Dio cles Eudemus

Euto cius of Aokolon Hero Nicomedes

Pappus Plato

Pro clus Protagoras Simplicius

Theon of Smyrna

Figure 1. Ma jor Commentators on Geometry

are outside our story other than to say conventional wisdom would have said that the constructionist

view was not Euclid's.

The received story of Euclid is a jumble of comments collected in many contexts. To hear

the philosophers tell it, Euclid was as much a philosopher as mathematician. Many logicians talked

of a grand scheme by Euclid to set the foundations of mathematics and mathematical logic. But

was this his goal? And what was the intellectual climate preceding and during Euclid's time even

though we do not know precisely when that was? There is confusion over what commentators have

read into the text and what the ancient geometers might themselves have b elieved. Our problem

was to understand the \true" Euclid.

Section 2 lo oks at the men and their times. Section 3 lo oks at the text of the Elements itself.

What might b e Euclid's view of how geometry is conducted in Section 4.

2. The Ancients

There seem to be far more ancient commentators Fig. 1 on geometry than there are

investigators Fig 2. The purp ose of this section is to identify those p eople who probably had a

direct in uence on the technical content of the Elements. 3

Mid 5th Century to Mid 4th Century

Hipp o crates of Chios Oenipides of Chios of Cyrene

Time of Plato and Aristotle

Archytas of Tarentum

Leo damus

Theaetetus of Athens

Post Euclid

App olonius

Figure 2. Ma jor Contributors to Geometry

2.1. The Men Themselves. Of all those mentioned in the history of geometry there are four

pre-Euclideans whom I will mention here. There are two groups: one in the 4th Century and the

other in the time of Plato in the 3rd Century.

2.1.1. 4th Century Investigators. In the 4th century BC, Hipp o crates and gure promi-

nently in the century preceding the establishment of Plato's Academy. Oenopides is reputed to b e

the p erson who required only rulers and compasses b e used. In several places, Knorr [9 ] expresses

doubt that such a requirement was ever made. It seems highly unlikely that the early geometers

put any arti cial imp ediments in the way of development[9,p. 345]. Hipp o crates was, apparently,

the geometer of the 4th century. He added signi cantly to the knowledge of his time, but he was

also considered the rst compiler of geometry [7, p. 179]. 4

2.1.2. Plato's Academy. After Hipp o crates and Oenopides there is a lull in geometry research but

there is also the founding of Plato's Academy. Plato himself was not a mathematician and the

Academywas not a hotb ed of mathematical research. Plato is known to have commented that the

metho ds of geometry were not abstract enough; several centuries later, Carpus would complain that

the metho ds were to o abstract to b e useful [9 , p. 364{365]. The Academy did make a signi cant

turn towards mathematics when Eudoxus joined it.

Eudoxus. Eudoxus is most rememb ered for his work on approximations to limits, or the

so-called \metho d of exhaustion." He was also a consummate geometer. How much did he know

ab out constructive metho ds? There is a group of historians that place heavy emphasis on religious

motives for solving geometric problems. Seidenb erg [11 , 12 ] calls these \p eg and cord" metho ds.

These techniques were known throughout Western Asia. The story is that the priests of various

religions solved several imp ortant problems constructively and the Greek mathematicians to ok it

from there. Interestingly enough, Eudoxus did sp end time in Egypt studying astronomy with the

priests. The apparent impredication in Theorem 1, Bo ok 1 can be seen di erently through the

metho ds discussed in Seidenb erg [11 , 12 ]. One need not invoke a continuity principle at all, but

simply observe the use of cord and peg metho ds. On the other hand, religion was probably just

another application [9 , p. 16].

Eudoxus [6] is given credit for developing deductive reasoning. He also seems to have made

a distinction b etween magnitudes and numb ers. While numb ers were prior and ruled by \common

notions", magnitudes were not. Magnitudes were ruled by ratios and prop ortions.

Menaechmus. Menaechmus was at the Academy at the same time as Eudoxus, probably as

a student, then as a teacher. Menaechmus puts the priorityto problems over theorems [9, p. 76,

p. 351]. There are twotyp es of problems [9 , p. 359]:

A. Those that lead to the determination by certain gures that result in another gure and

B. Those that lead to the construction of a gure with sp eci ed prop erties. 5

Name Dates

So crates 469 BC to 399 BC

Plato c. 428 BC to c. 347 BC

Eudoxus 408 BC to 355 BC

Aristotle 384 BC to 322 BC

Euclid No rm guesses: 325 BC to 300 BC

Archimedes 287 BC to 212 BC

Theon c. 335 AD to 395 AD

Pro clus 410 AD to 485 AD

Figure 3. Chronology of Imp ortant Greeks

Menaechmus seems to b e the rst recorded intuitionist fo cusing on establishing existence.

Euclid. Much controversy could be avoided if we knew more ab out Euclid the man. But

Euclid is an unknown gure. To omer [13 ] says \the biographical linking Euclid with Alexandria

and I are worthless references by late authors Pappus and Pro clus who seem to have

no more information than we do." Knorr is kinder [9 , p. 138]. \The biographical data on Euclid

is meager." Be that as it may, the dates Fig. 3 for the main characters put Euclid in some

interesting company.

As for Euclid himself, it is b etter to think of him [9 , p 138] as a \compiler and an e ective

teacher." His contributions were seminal to the development of problem solving in the 3rd Century

BC [9 , p 138]. All references seem to agree that Euclid was not himself a terri c geometer. Despite

this, one must wryly note that he did write the all time b est seller in geometry if not all of

mathematics. And this is a teaching text and not a research tract to b o ot!

The lack of information ab out Euclid leads to many apparently invented stories ab out

his philosophical leanings. I was taught he was a Platonist. Given the chronology in Fig. 3 one

might be inclined to accept that if one has some idea of Euclid's age when he did this work. 6

But, he also might well have known Aristotle and b een more oriented towards his scho ol. The

habit of later commentators putting their own spin in earlier events is all to o common. We have

all heard the tale of do omed to be shipwrecked for having divulged the secret of the

irrational to the uninitiated. \Wemust b e quite skeptical of placing much sto ckininterpretations

on unsubstantiated stories. [9 , p. 88]" We return to this in Section 4.

3. Euclid's Elements

Ultimately, our understanding rests on the Elements. There is the do cument itself and there

are the organization and metho ds of Euclid.

3.1. The Text. The b est precis of the Elements itself is,

\Yet for all its sophistication of its logical structure and intricacy of some of its construc-

tions, ::: , the Elements is predominantly a treatise of an intro ductory sort, as its name

implies. The researches indicated were initiated decades b efore Euclid by Hipp o crates,

Thaetetus and Eudoxus and advanced by their successors." [9 , p. 102]

For much of history, the recension of Theon [13 ] was the basis of knowledge ab out the Elements.

In 1814, a copy of the Elements was found in the Vatican library. What this means is that until

1814, all discussion ab out Euclid was really ab out Theon's annotated version. Clearly, Elements is

not an exhaustiveinventory of metho ds, problems, or results available to Euclid [9 , p. 92].

3.2. The Organization of the Elements. Because the text is essentially a textb o ok, Euclid's

formalized organization probably did not follow the order of discovery [9 , p. 71]. Given that the

results are not Euclid's p ersonal research and that he was writing a teaching do cument, we can

assume that he chose those pro ofs for their p edagogical imp ortance. This view is enhanced when

you carefully consider Bo ok I: Theorem 22 starts afresh from rst principles.

A Note on Notes. The convention to talking ab out theorems in Euclid is to quote the book rst

and theorem numb er second. For example, I.22 is theorem 22 in Bo ok I. 7

It is useful to lo ok at its organization.

 De nitions. De nitions in Bo ok I are not formalized as in later texts. There is not much of

interest here except that Euclid omitted ab out as many de nitions as he gave.

 Problems. Problems are cast with an in nitive: \to nd" or \to construct". A problem refers

to the pro duction of a sp eci c gure from sp eci c conditions. There are decided advantages

to a problem approach [9 , p.348{354]. Lo cus problems arise when there are multiple solutions.

 Postulates. There are three geometric p ostulates. These are stated in the form of problems

and not theorems. That is, the p ostulates tell us that certain gures can be found given

certain conditions and not merely that solutions \exist." The clear implication here is that

the p ostulates were minimal with resp ect to solving problems: we need not concern ourselves

with \simpler" problems. These p ostulates make for an ordering of problems [9, p. 350].

Before continuing, let us be clear on this point: there are no axioms in Euclid and the

postulates are not stated as relations but as algorithms!

 Common Notions. Common notions are what wewould call axioms and are related to theorems

and statements/pro ofs [7 , p. 169]. It is not clear that the p ostulates as translated t in this

category or not. There app ears to be quite a bit of doubt of as to what Euclid actually

included where [5 , 195{240]. We note also that the mo dern concept of axiom comes much

later. In Euclid's time, common notions were made for veri cation testing purp oses [6]. In

view of the comment ab out p ostulates, axioms allow for the ordered sequence of theorems.

 Theorems. Theorems are cast as a conditional assertion relating to a sp eci ed con guration.

They refer to general classes of ob jects. Theorems are the appropriate mo de when one is

concerned with formal organization of known results, rather than with the discovery of what

is yet unknown [9 ]. For example, I.20 is a theorem, not a problem although it seems more

naturally stated as a problem. 8

 Analysis. In analysis, we \pretend" to have the gure and reason backwards towards axioms

or previous theorems [9 , p. 354{360]. Aristotle holds that analysis is deductively admissible

[9 , p. 75].

 Synthesis. The desired gure is constructed by starting with the hyp otheses and moving

forward deductively to the conclusion [9, p. 354{360].

 Diorisms. Diorisms were formalized by the mid-4th century BC. Diorisms give auxiliary

conditions which, when supplied, guarantee that a problem is solvable [9, p. 358]. Thus

diorisms allow analysis to b e converted to synthesis.

3.3. The Metho ds of the Elements. The standard metho d calls for analysis rst, then synthesis

[9 , p. 9]. From Hero's comments on Metrica [9, fn 87, p. 376 ]: \The ob jective is to derive a

computational rule or pro cedure by means of what is called an `analysis' and then in the following

`synthesis' to work out a solution in particular numb ers via the derived rule." The editorial style

of the commentators seems to prefer the synthetic treatment but this obscures the essential lines

of thought[9,p. 9].

The Elements is ab out nothing if it is not ab out justi cation.

\In dialectics the grounds of knowledge b ecome a central issue. What is true? How

do es one know what is true and distinguish from the false? How do es one communi-

cate and teach? ::: . But surely this general environment had an equivalent impact on

mathematics. Examining their arithmetic and geometric techniques, they b egan to seek

justi cations ::: . Once the quest for justi cation was underway, the nature of math-

ematics itself would lead to deductive forms ::: . Furthermore, the incentive to teach

spurred e orts to organize large areas of mathematics into coherent systems." [7, p. 179]

However, this sheds no light on what Euclid thought was sucient justi cation. Surely Euclid

would ob ject to hiding the lines of development from his students: after all, as a teacher he would 9

want his students to understand how the constructions and theorems come ab out. We turn to this

in Section 4.

3.3.1. The Theorems versus Problems Debate. The historical record indicates that ancient com-

mentators were often tainted by their own philosophical p ositions: Pro clus is a good example.

These predisp ositions obscure the debate over the primacy of theorems over problems. We seem to

have more information on the philosophers' p ositions than the geometers' p ositions.

Pro clus observed that Bo ok I is evenly split b etween theorems and problems. \In this way,

the problems often serve as justi cation for the intro duction of auxiliary terms in later prop ositions"

[9 , p. 350]. Problems and theorems are interchangeable. Perhaps more imp ortant is the idea that

successful solution to problems leads to theorems and vice versa. In my own exp erience, this

relationship is crucial, as the insight needed to prove a theorem is often buried in the construction.

Amphimonous, who is linked to Plato's successor and nephew Sp eusippus, is said to have held to

the view that everything should b e stated as a theorem [9, p. 75]. \Through all of this, the technical

literature does not suggest that everything should be in one form: theorem exclusive or problem."

[Italics mine] [9 , fn 60, p. 374].

3.3.2. The Links Between The Elements and Philosophy. Euclid's Elements and Aristotle's Poste-

rior Analytics represent the state of the art at the time for geometry and philosophy resp ectively

[7 , p. 164{165]. \In this way, Aristotle's discussion of the rst principles of pro of b ecomes a

straight-forward commentary on the actual pro cedures of pro of employed by Euclid's immediate

predecessors [7 , p. 166{167]. \Indeed, judging from the epistemological views of Plato and Aristo-

tle, one cannot escap e the conviction that the in uence of mathematics on philosophywas far more

signi cant than any in uence in the converse direction" [7 , p. 179]. On the other hand, McKirahan

seems not so sure [10 ]. 10

4. A Possible Philosophy of Geometry

The historical study enables us to at least conjecture what the geometers thought ab out how

the study of geometry should b e conducted. A naive reading of the commentaries of Euclid would

lead one to think that there was much philosophical input on the conduct of geometric inquiries.

Eudoxus seems to b e the only ancient geometer who was interested in formal considerations. More

imp ortantly,

\Thus there app ears to b e no dialectical motive b ehind Euclid's statement of p ostulates

or his presentation of geometric construction. Moreover, in setting out the p ostulates,

he did not aim to restrict the whole eld of research on constructions to those which

can be e ected in practice or even to suggest that these means were somehow more

privileged among the variety of construction devices p ossible[example: cub e duplication

by non-Euclidean means]." [7]

Knorr p oints out that virtually any philosophical system, ancient or mo dern, is supp orted [8 , p.

141] by the Elements. This means that one could take a philosophical p oint of view, like construc-

tionism and claim that Euclid used that view to develop the Elements. This would seem to t

Pro clus nicely. The commentators did help maintain information ab out \metageometry" issues by

discussing certain issues. Therefore, it is likely that the geometers themselves had no commitment

to philosophical p ositions in geometry. The technical treatises are devoid of philosophical concerns

[8 , p. 140{141].

4.1. What Did Geometers Think Ab out. If not driven by philosophical ideas, what did the

geometers concern themselves with in their investigations? A short list:

A. How should the eld b e partitioned by problem and solution metho dology?

B. What is the role of problems vis- a-vis theorems?

C. What is the status of analysis vis- a-vis synthesis? 11

D. What conditions should b e imp osed on which allowable technologies?

E. What are the judgments for accepting a solution that has b een found to a problem?

It should b e noted that almost exactly these same questions are b eing asked in the reformation of

calculus.

For our purp oses, a small numb er of questions need to b e addressed in detail.

4.1.1. Superposition. A fundamental approach to the computable theory rests on sup erp osition.

The formalist commentators hold that Euclid used it reluctantly. Knorr investigates sup erp osition

in detail [7 , p. 159{161] and concludes that Euclid probably did not shun the concept but rather

found it was not needed b ecause of the order of presentations.

4.1.2. Axiomatization. While axioms play a ma jor role in mathematics to day, they seemed not to

b e so imp ortant in ancient times. Ihave made a case that the technical treatises do not delveinto

subtle questions of axiomatics [9, p. 8]. But what were the ancient mathematical views?

A. \The notion that ancient mathematics was somehowavast exercise in dialectical philosophy

must miss a very imp ortant p oint: that geometry is ro oted in an essentially practical enterprise

of problem solving." [9 , p. 11]

B. \The subsequent indicates that the success of axiomatizing e ort [by

the 3rd century] eventually served to discourage the creative forms of research which could

have advanced mathematical knowledge." [7 , p. 178]

To develop the constructionist viewp oint, we replace the concept that axioms are logical state-

ments expressing fundamental conditions of the system by primitive computational capabilities

that achieve the same goal. Once the computational capabilities are asso ciated with simple rela-

tions we can expand the relations into logical statements.

\By emphasizing the solution of problems the ancients do not necessarily intend physical

constructions, although the literature includes many examples of the typ e. But neither 12

do they emphasize the pure asp ects of theory to the exclusion of interests in the practical

asp ects of their results. Even in these advanced theoretical e orts in geometry, the

ancients are still sensitive, if only in an indirect way, to the demands of practice. [8, p.

140]"

Thus, the ancients would probably accept Brouwer's view: mathematics is done with mental con-

structions. The ancient geometers would basically approve of a constructive formulation that

emphasizes idealization over axiomatization.

4.2. Restrictions to Ruler and Compasses. Oenipides is blamed for this restriction, but I have

already made a case against the view in the section ab ove. The restriction to ruler and compass,

then, should be seen as a p edagogical one. It is clear that much of geometry dep ends on the

ruler and the compass. This might lead a philosophically oriented ancient geometer to formulate

aversion of Church's Thesis: Anything that is Euclidean is constructible by a ruler and compass.

We might ask, for example, if there are simpler pro cedures for ruler only, say which pro duce

sub-theories? As wewell know, the answer is yes [3]. In light of mo dern studies, the restriction is

achallenge to action, not a prohibition. We also know that ruler and compasses are not enough to

solve all problems p osable in geometry, leaving op en the question of what is an adequate base set

of pro cedures.

4.3. The Balance between Practical and Theoretical. There has b een a tendency in recent

years to see mathematics as detached from its application. You might call this the pure versus

applied debate. In my opinion, this is not even a sensible question, since elements of mathematics

and other sciences as well can serve in b oth capacities; clearly, pure drives applied and vice versa.

The history of mathematics shows that formal structure can slow down the search for so-

lutions but sometimes demand great sub ject insight [9, p. 76]. While we give great credit to

Euclid for presenting formalized metho ds, it is not until App olonius that the sub ject matter gets

formalized into formal metho ds of problem solving [9 , p. 346]. 13

Mathematics and science b ounce between two extremes: On the one hand we have, \The

mo dern mathematician may b e remote from such practical applications of his researches, but this

is clearly not the case with the ancients. [8 , p. 139]". The 20th century shows \developments in

analysis, geometry,numb er theory, and the theory of sets revealed that certain formal issues, long

ignored or unp erceived lay at the heart of imp ortant technical diculties." [9 , p. 87]

4.4. What Do We Conclude?

\Toevery thing there is a season, and a time to every purp ose under heaven" Ecclesiastes 3:1

What would ancient researchers say if we asked, \What are the principles that guide the

investigation of geometry?" I think they might answer something like this:

\We study and solve problems in a general sense. There are problems that we can solve

with our current metho ds and there are problems we cannot solve|or at least don't

knowhow to solve with the current metho ds. We use problems to motivate our research.

There are general theorems ab out geometry but wehave no catalog of all knowable facts

ab out what you call .

\As to formalization, wehave no prescrib ed set of concepts nor do we attempt to ban

concepts. Problems are approached by one set of techniques, and theorems by another,

but they seem to be inter-related. There is some discussion as to how these are inter-

related by analysis and synthesis. The ultimate question is, `How do we know we are

right?' Problems are the life blo o d of anywork."

\To paraphrase Wilbur Knorr, `You assume that the hallmark of our work is our

organization, tight structures of deductive reasoning and fully justi ed problem solutions'

[9 , p. 7]. No, we're just like you ::: . We put it down and re ne and argue over the

correctness of our thoughts. Wehave our go o d days and our bad days." 14

5. What Does This Have To Do With Computer Science

The explosive growth of computer technology and the numb er of p eople \practicing" com-

puter science is a historical rst. The Information Age and the Information Revolution have

spawned tremendous changes in science, mathematics, and a host of so ciological pro cess in busi-

ness, warfare, and home life. We will not b e in a p osition to taketwenty-three centuries to develop

computer science in the \slow" way mathematics develop ed. But we can hop e that computer sci-

ence is as go o d an acorn as geometry has b een for the oak tree of mathematics. Euclid do es sp eak

to computer science, though.

5.1. Some Observations.

A. It is clear that practical problems are interesting. The \Traveling Salesp erson Problem" is

such an example. What is allowable technology compass and ruler only argument has not

b een a problem b ecause the chip manufacturers develop more and faster chips.

B. Forcing arti cial restrictions on metho d is to b e avoided. Perhaps the ancients had an unfair

advantage: they invented the academic system but didn't have to live in it. One arti cial

restriction is that computation is exactly an axiomatic structure. Fetzer has convincingly

countered this argument in [4 ], to the howls of the computer science establishment.

Another asp ect of this problem of constraints has b een to demand that computer science fol-

low mathematical epistemology of pro of by derivation along mathematical logical structures.

I characterize this problem as the \Hammer Problem." A long standing joke in engineer-

ing says, \To a engineer with a hammer the whole world lo oks like a nail." To engineers,

mathematicians, and scientists with their hammers, computer science lo oks like a nail.

C. The interchangeability of problems and theorems is an interesting extension. An intriguing

asp ect is how to link them formally | a constructivist's goal. 15

D. Formalizing the analysis and synthesis asp ects of computing would be an imp ortant step

towards e ective teaching of concepts and generating generations of comp etent computer

scientists. By the way,Ihave not one clue as to how to translate the concept of diorisms.

5.2. Comments on Metho d. My original goal was to understand Euclid in terms of his metho d

with a \Hammer" of constructivism not Platonism. What can I say ab out that?

Because the p ostulates are stated as problems, I can at least conclude that the idea of

construction is prior to theorem. This is not completely surprising since systems of axioms must

be veri ed for consistency. Also interesting is the observation that the reasoning rules were for

veri cation, not derivation.

From Hero's comments in Section 3.3, we can conclude something ab out how \metho ds"

were viewed. The goal was to derived a computational rule that linked analysis to synthesis. In

this, apparently, the ancients say synthesis as the goal | just like programming.

To close this section, geometry fairly well shouts advice ab out formalization: \The subse-

quent history of mathematics indicates that the success of axiomatizing e ort [by the 3rd century]

eventually served to discourage the creative forms of research which could have advanced mathe-

matical knowledge" [7 , p. 178]. Computer science is full of e orts to over-formalize everything. We

seem not to have learned much through history.

5.3. So What? Iwould prop ose that a goal of academicians should b e to come up with Euclid's

Elements for computer science: a catalog of all the metho ds and problems that we can solve.

Teaching computer science might far more rewarding and less magical if we had such a text. After

all, it nourished twenty-three hundred years of mathematicians.

It is a pity we no longer teach it in our scho ols and computer scientists are ignorant of its

legacy. 16

Acknowledgments

My thanks to Professor Stuart Silvers, chair of the Philosophy Department at Clemson,

for listening to me rave on ab out this pro ject and his subsequent comments on the manuscript.

Professor J. L. Hirst of Appalachian State University provided manyvaluable comments on presen-

tation and various logical concepts. Professors S. T. Hedetniemi of Clemson University and Rob ert

Thomas of the University of Manitoba also provided many helpful suggestions.

The historical and philosophical asp ects of this work are certainly not mine alone. Many

thanks to Professors Elizab eth Carney and Alan Sha er of the Clemson University History Depart-

ment for the initial entries to the historical literature. Dr. G. L. To omer of Harvard suggested I

contact Professor Wilbur Knorr at Stanford. Professor Rob ert Thomas of the University of Mani-

toba was of considerable help, guiding me toward Professor R. D. McKirahan who suggested some

of the connections with Posterior Analytics and Dr. W. Anglin of the UniversityofToronto.

Finally, Professor H. R. Mendell was of considerable help at a sad time for him. I had mailed

a letter to Professor Knorr that arrived just three days b efore his untimely death due to melanoma

at the age of 51. Dr. Mendell stepp ed in to help me, suggesting a long list of texts and articles to

read. This pap er is largely based on sources recommended by Professor Mendell.

Lastly, as I read Professor Knorr's texts, I b ecame increasing sorry I never knew him. I hop e

I've b een a go o d interpreter of the evidence.

References

[1] Errett Bishop and Douglas Bridges. Constructive Analysis. Springer-Verlag, 1985.

[2] Carl B. Boyer. A History of Mathematics. Wiley, 1989.

[3] H.S.M. Coxeter. Introduction to Geometry. John Wiley and Sons, Inc., New York, 2 edition, 1969.

[4] James H. Fetzer. Program veri cation: the very idea. CACM, 319, 1988.

[5] Sir Thomas L. Heath. The Thirteen Books of Euclid's Elements, volume I{I I I. Dover Publications, Inc., New

York, 1956. 17

[6] Morris Kline. Mathematical Thought from Ancient to Modern Times. Oxford University Press, 1972.

[7] Wilbur R. Knorr. In nity and continuity: The interaction of mathematics and philosophyinantiquity. In Norman

Ketzmann, editor, In nity and Continuity in Ancient and Medieval Thought, pages 146{164. Cornell University

Press, 1981.

[8] Wilbur R. Knorr. Construction as existence pro of in ancient geometry. Ancient Philosophy, pages 125{148, 1983.

[9] Wilbur R. Knorr. The Ancient Tradition of Geometric Problems. Birkhaser, 1986.

[10] Richard D. McKirahan. Principles and Proofs: Aristotle's Theory of Demonstrative Science. Princeton University

Press, 1992.

[11] A. Seidenb erg. Peg and CordinAncient Greek Geometry,volume 24. Scripta Mathematica, 1959.

[12] A. Seidenb erg. The ritual origin of geometry. History of the Exact Sciences, 1:487{527, 1963.

[13] G. L. To omer. Euclid. In Oxford Classical Dictionary, page 564. Oxford University Press, 1996.

[14] Walter P. van Stigt. Brouwer's Intuitionism, chapter 2{4. Elsevier Science Publishing Company, Inc., 1990.

QA29.B697S25.

[15] Jan von Plato. The axioms of constructive geometry. Annals of Pure and AppliedLogic, 76:169{200, 1995.

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